Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 1 (2012), Pages 190-196.
NEW INEQUALITIES USING FRACTIONAL Q-INTEGRALS
THEORY
(COMMUNICATED BY R.K. RAINA)
ZOUBIR DAHMANI, ABDELKADER BENZIDANE
Abstract. The aim of the present paper is to establish some new fractional
q-integral inequalities on the specific time scale: Tt0 = {t : t = t0 q n , n ∈
N } ∪ {0}, where t0 ∈ R, and 0 < q < 1.
1. Introduction
The integral inequalities play a fundamental role in the theory of differential
equations. Significant development in this area has been achieved for the last two
decades. For details, we refer to [12, 13, 16, 22, 18, 19] and the references therein.
Moreover, the study of the the fractional q-integral inequalities is also of great
importance. We refer the reader to [3, 15] for further information and applications.
Now we shall introduce some important results that have motivated our work. We
begin by [14], where Ngo et al. proved that for any positive continuous function f
on [0, 1] satisfying
∫ 1
∫ 1
f (τ )dτ ≥
τ dτ, x ∈ [0, 1],
x
x
and for δ > 0, the inequalities
∫
∫
1
1
f δ+1 (τ )dτ ≥
0
and
∫
τ δ f (τ )dτ
(1.1)
τ f δ (τ )dτ
(1.2)
0
∫
1
f
δ+1
1
(τ )dτ ≥
0
0
hold.
Then [10], W.J. Liu, G.S. Cheng and C.C. Li established the following result:
∫
∫
b
b
f α+β (τ )dτ ≥
a
(τ − a)α f β (τ )dτ,
a
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C50.
Key words and phrases. Fractional q-calculus, q-Integral inequalities.
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted November 24, 2011. Published February 20, 2012.
190
(1.3)
FRACTIONAL QUANTUM INTEGRAL INEQUALITIES
191
where α > 0, β > 0 and f is a positive continuous function on [a, b] such that
∫ b
∫ b
f γ (τ )dτ ≥
(τ − a)γ dτ ; γ := min(1, β), x ∈ [a, b].
x
x
Recently, Liu et al. [11] proved that for any positive, continuous and decreasing
function f on [a, b], the inequality
∫b β
∫b
f (τ )dτ
(τ − a)δ f β (τ )dτ
a
≥ ∫ab
, β ≥ γ > 0, δ > 0
(1.4)
∫b
f γ (τ )dτ
(τ − a)δ f γ (τ )dτ
a
a
is valid.
This result was generalized to the following [11]:
Theorem 1.1. Let f ≥ 0, g ≥ 0 be two continuous functions on [a, b], such that f
is decreasing and g is increasing. Then for all β ≥ γ > 0, δ > 0,
∫b β
∫b δ
f (τ )dτ
g (τ )f β (τ )dτ
a
≥ ∫ab
.
(1.5)
∫b
f γ (τ )dτ
g δ (τ )f γ (τ )dτ
a
a
The same authors established the following result:
Theorem 1.2. Let f ≥ 0 and g ≥ 0 be two continuous functions on [a, b] satisfying
(
)(
)
f δ (τ )g δ (ρ) − f δ (ρ)g δ (τ ) f β−γ (τ ) − f β−γ (ρ) ≥ 0; τ, ρ ∈ [a, b].
Then, for all β ≥ γ > 0, δ > 0 we have
∫ b δ+β
∫b δ
f
(τ )dτ
g (τ )f β (τ )dτ
a
≥ ∫ab
.
∫b
f δ+γ (τ )dτ
g δ (τ )f γ (τ )dτ
a
a
(1.6)
More recently, using fractional integration theory, Z. Dahmani et al. [6, 7] established some new generalizations for [11].
Many researchers have given considerable attention to (1),(3) and (4) and a number
of extensions and generalizations appeared in the literature (e.g. [4, 5, 6, 7, 9, 10]).
The purpose of this paper is to derive some new inequalities on the specific time
scales Tt0 = {t : t = t0 q n , n ∈ N } ∪ {0}, where t0 ∈ R, and 0 < q < 1. Our
results, given in section 3, have some relationships with those obtained in [11] and
mentioned above.
2. Notations and Preliminaries
In this section, we provide a summary of the mathematical notations and definitions used in this paper. For more details, one can consult [1, 2].
Let t0 ∈ R. We define
Tt0 := {t : t = t0 q n , n ∈ N } ∪ {0}, 0 < q < 1.
(2.1)
For a function f : Tt0 → R, the ∇ q-derivative of f is:
∇q f (t) =
f (qt) − f (t)
(q − 1)t
(2.2)
192
Z. DAHMANI, A. BENZIDANE
for all t ∈ T \ {0} and its ∇q-integral is defined by:
∫
t
f (τ )∇τ = (1 − q)t
0
∞
∑
q i f (tq i )
(2.3)
i=0
The fundamental theorem of calculus applies to the q-derivative and q-integral. In
particular, we have:
∫ t
f (τ )∇τ = f (t).
(2.4)
∇q
0
If f is continuous at 0, then
∫
t
∇q f (τ )∇τ = f (t) − f (0).
(2.5)
0
Let Tt1 , Tt2 denote two time scales and let f : Tt1 → R be continuous, and g : Tt1 →
Tt2 be q-differentiable, strictly increasing such that g(0) = 0. Then for b ∈ Tt1 , we
have:
∫ b
∫ g(b)
f (t)∇q g(t)∇t =
(f ◦ g −1 )(s)∇s.
(2.6)
0
0
The q-factorial function is defined as follows:
If n is a positive integer, then
(t − s)(n) = (t − s)(t − qs)(t − q 2 s)...(t − q n−1 s).
(2.7)
If n is not a positive integer, then
∞
∏
1 − ( st )q k
.
1 − ( st )q n+k
(t − s)(n) = tn
(2.8)
k=0
The q-derivative of the q-factorial function with respect to t is
∇q (t − s)(n) =
1 − qn
(t − s)(n−1) ,
1−q
(2.9)
and the q-derivative of the q-factorial function with respect to s is
∇q (t − s)(n) = −
1 − qn
(t − qs)(n−1) .
1−q
(2.10)
The q-exponential function is defined as
eq (t) =
∞
∏
(1 − q k t), eq (0) = 1
(2.11)
k=0
The fractional q-integral operator of order α ≥ 0, for a function f is defined as
∫t
1
α−1
∇−α
f (τ )∇τ ; α > 0, t > 0,
(2.12)
q f (t) = Γq (α) 0 (t − qτ )
where Γq (α) :=
1
1−q
∫1
0
u α−1
)
eq (qu)∇u.
( 1−q
FRACTIONAL QUANTUM INTEGRAL INEQUALITIES
193
3. Main Results
In this section, we state our results and we give their proofs.
Theorem 3.1. Suppose that f is a positive, continuous and decreasing function on
Tt0 . Then for all α > 0, β ≥ γ > 0, δ > 0, we have
β
∇−α
q [f (t)]
γ
∇−α
q [f (t)]
≥
δ β
∇−α
q [t f (t)]
δ γ
∇−α
q [t f (t)]
, t > 0.
(3.1)
Proof. For any t ∈ Tt0 then for all β ≥ γ > 0, δ > 0, τ, ρ ∈ (0, t), we have
(
)(
)
ρδ − τ δ f β−γ (τ ) − f β−γ (ρ) ≥ 0.
(3.2)
Let us consider
(
)(
)
H(τ, ρ) := f γ (τ )f γ (ρ) ρδ − τ δ f β−γ (τ ) − f β−γ (ρ) .
(3.3)
Hence, we can write
∫ t∫ t
(t − qτ )(α−1) (t − qρ)(α−1)
−1
β
−α δ γ
2
H(τ, ρ)∇τ ∇ρ = ∇−α
q [f (t)]∇q [t f (t)]
Γq (α)
Γq (α)
0
0
γ
−α δ β
−∇−α
q [f (t)]∇q [t f (t)] ≥ 0.
(3.4)
The proof of Theorem 3.1 is complete.
We have also the following result:
Theorem 3.2. Let f, g and h be positive and continuous functions on Tt0 , such
that
(
)( f (ρ) f (τ ) )
g(τ ) − g(ρ)
−
≥ 0; τ, ρ ∈ (0, t), t > 0.
(3.5)
h(ρ) h(τ )
Then we have
∇−α
∇−α
q (f (t))
q (gf (t))
(3.6)
≥
,
−α
∇q (h(t))
∇−α
q (gh(t))
for any α > 0, t > 0.
Proof. Let f, g and h be three positive and continuous functions on Tt0 . By
(3.5), we can write
g(τ )
f (ρ)
f (τ )
f (ρ)
f (τ )
+ g(ρ)
− g(ρ)
− g(τ )
≥ 0,
h(ρ)
h(τ )
h(ρ)
h(τ )
(3.7)
where τ, ρ ∈ (0, t), t > 0.
Therefore,
g(τ )f (ρ)h(τ ) + g(ρ)f (τ )h(ρ) − g(ρ)f (ρ)h(τ ) − g(τ )f (τ )h(ρ) ≥ 0, τ, ρ ∈ (0, t), t > 0.
(3.8)
(α−1)
)
Multiplying both sides of (3.8) by (t−qτ
Γq (α)
equality with respect to τ over (0, t), yields
, then integrating the resulting in-
−α
−α
−α
f (ρ)∇−α
q gh(t) + g(ρ)h(ρ)∇q f (t) − g(ρ)f (ρ)∇q h(t) − h(ρ)∇q gf (t) ≥ 0,
(3.9)
and so,
−α
−α
−α
∇−α
(3.10)
q f (t)∇q gh(t) − ∇q h(t)∇q gf (t) ≥ 0.
194
Z. DAHMANI, A. BENZIDANE
This ends the proof of Theorem 3.2.
Using two fractional parameters, we have a more general result:
Theorem 3.3. Let f, g and h be be positive and continuous functions on Tt0 , such
that
(
)( f (ρ) f (τ ) )
g(τ ) − g(ρ)
−
≥ 0; τ, ρ ∈ (0, t), t > 0.
(3.11)
h(ρ) h(τ )
Then the inequality
−α
−ω
−ω
∇−α
q (f (t))∇q (gh(t)) + ∇q (f (t))∇q (gh(t))
−ω
−ω
−α
∇−α
q (h(t))∇q (gf (t)) + ∇q (h(t))∇q (gf (t))
holds, for all α > 0, ω, t > 0.
≥1
Proof. As before, from (3.9), we can write
(t − qρ)ω−1 (
−α
f (ρ)∇−α
q gh(t) + g(ρ)h(ρ)∇q f (t)
Γq (ω)
)
−α
−g(ρ)f (ρ)∇−α
h(t)
−
h(ρ)∇
gf
(t)
≥ 0,
q
q
(3.12)
(3.13)
which implies that
−α
−α
−ω
∇−ω
q (f (t))∇q (gh(t)) + ∇q (f (t))∇q (gh(t))
−ω
−ω
−α
≥ ∇−α
q (h(t))∇q (gf (t)) + ∇q (h(t))∇q (gf (t)).
Theorem 3.3 is thus proved.
(3.14)
Remark 3.4. It is clear that Theorem 3.2 would follow as a special case of Theorem 3.3 for α = ω.
We further have
Theorem 3.5. Suppose that f and h are two positive continuous functions such
that f ≤ h on Tt0 . If fh is decreasing and f is increasing on Tt0 , then for any
p ≥ 1, α > 0, t > 0, the inequality
∇−α
q (f (t))
∇−α
q (h(t))
≥
p
∇−α
q (f (t))
(3.15)
p
∇−α
q (h (t))
holds.
Proof. Thanks to Theorem 3.2, we have
∇−α
q (f (t))
≥
p−1
∇−α
(t))
q (f f
p−1 (t))
∇−α
∇−α
q (h(t))
q (hf
The hypothesis f ≤ h on Tt0 implies that
.
(t − qτ )α−1 p−1
(t − qτ )α−1 p
hf
(τ ) ≤
h (τ ), τ ∈ (0, t), t > 0.
Γq (α)
Γq (α)
(3.16)
(3.17)
Then by integration over (0, t), we get
p−1
p
∇−α
(t)) ≤ ∇−α
q (hf
q (h (t)),
(3.18)
FRACTIONAL QUANTUM INTEGRAL INEQUALITIES
195
and so,
p−1
∇−α
(t))
q (f f
≥
p
∇−α
q (f (t))
p−1 (t))
p
∇−α
∇−α
q (hf
q (h (t))
Then thanks to (3.16) and (3.19), we obtain (3.15).
(3.19)
.
Another result is given by the following theorem:
Theorem 3.6. Suppose that f and h are two positive continuous functions such
that f ≤ h on Tt0 . If fh is decreasing and f is increasing on Tt0 , then for any
p ≥ 1, α > 0, ω > 0, t > 0, we have
−α p
−ω p
−ω
∇−α
q (f (t))∇q (h (t)) + ∇q (f (t))∇q (h (t))
−ω
−ω
−α p
p
∇−α
q (h(t))∇q (f (t)) + ∇q (h(t))∇q (f (t))
≥ 1.
(3.20)
Proof. We take g := f p−1 in Theorem 3.5. Then we obtain
−ω
p−1
−α
p−1
∇−α
(t)) + ∇−ω
(t))
q (f (t))∇q (hf
q (f (t))∇q (hf
−ω
−ω
−α p
p
∇−α
q (h(t))∇q (f (t)) + ∇q (h(t))∇q (f (t))
The hypothesis f ≤ h on Tt0 implies that
≥ 1.
(t − qρ)ω−1 p−1
(t − qρ)ω−1 p
hf
(ρ) ≤
h (ρ), ρ ∈ (0, t), t > 0.
Γq (ω)
Γq (ω)
(3.21)
(3.22)
Integrating both sides of (3.22) with respect to ρ over (0, t), we obtain
p−1
p
∇−ω
(t)) ≤ ∇−ω
q (hf
q (h (t)).
(3.23)
Hence by (3.18) and (3.23), we have
−ω
p−1
−α
p−1
∇−α
(t)) + ∇−ω
(t))
q f (t)∇q (hf
q f (t)∇q (hf
−α p
−ω
−ω p
≤ ∇−α
q f (t)∇q (h (t)) + ∇q f (t)∇q (h (t)).
By (3.21) and (3.24), we complete the proof of this theorem.
(3.24)
Remark 3.7. Applying Theorem 3.6, for α = ω, we obtain Theorem 3.5.
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Z. Dahmani
Laboratory of Pure and Applied Mathematics, LPAM, Faculty SEI, UMAB, University
of Mostaganem
E-mail address: zzdahmani@yahoo.fr
A. Benzidane
Department of Mathematics, Faculty of Exact Sciences, UMAB University of Mostaganem
E-mail address: aekm27@yahoo.fr